We implemented the VDF equations of motion, Eqs. (REF ) and (), in the hadronic transport code SMASH [1]}, version 1.8 [2]}, where simulating hadronic non-equilibrium dynamics is achieved through numerically solving the Boltzmann equation, in this context often also called the Vlasov equation, the Boltzmann-Uehling-Uhlenbeck (BUU) equation, or the Vlasov-Uehling-Uhlenbeck (VUU) equation. The specification comes from solving the Boltzmann equation for the time evolution of the phase-space density \(f(t, {x},{p})\)  in the presence of the mean-field \(U({x},{p})\) ,
\(\bigg [ \frac{\partial {}}{\partial {t}} + \frac{\partial {H_{(1)}}}{\partial {{p}}} {\nabla }_{{p}} - \frac{\partial {H_{(1)}}}{\partial {{x}}} {\nabla }_{{p}} \bigg ] f(t, {x},{p}) = I_{\textrm {coll}}~,\) 
